Question

A parallel plate capacitor with circular plates of radius $R$ is being charged as shown in the figure. Let $B(r)$ be the induced magnetic field at a distance $r$ from the central axis between the plates. Assuming $d \ll R$, the ratio $\frac{B(2d)}{B(d)}$, while charging, is

• A. 2
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. 1

Hint:

Inside the plates ($r \lt R$), the induced magnetic field grows linearly with $r$ ($B \propto r$).
Outside the plates ($r \gt R$), it falls off as $1/r$ ($B \propto 1/r$).
Always check if the given distances are inside or outside the radius $R$!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem asks for the ratio of the induced magnetic fields at two different distances from the central axis of a circular parallel plate capacitor during its charging phase.

Step 2: Key Formula or Approach:
According to the Ampere-Maxwell Law, the magnetic field induced inside the plates of a capacitor at a distance $r \lt R$ is:
\[ \oint B \cdot dl = \mu_0 I_{\text{enclosed}} \]
Since there is no real conduction current between the plates, the magnetic field is created by the displacement current $I_d$:
\[ B(r) \cdot 2\pi r = \mu_0 I_d \left( \frac{\pi r^2}{\pi R^2} \right) \implies B(r) \propto r \quad (\text{for } r \lt R) \]

Step 3: Detailed Explanation:

• The capacitor has circular plates of radius $R$.

• We are given the condition $d \ll R$. This means both $d$ and $2d$ are much smaller than the radius $R$.
- Therefore, both points are located inside the region between the plates ($r \lt R$).

• Since both points lie inside the capacitor plates, the magnetic field at both locations is directly proportional to the distance from the central axis:
\[ B(r) = \left( \frac{\mu_0 I_d}{2\pi R^2} \right) r \]

• Let us find the ratio of the magnetic field at $r = 2d$ to that at $r = d$:
\[ \frac{B(2d)}{B(d)} = \frac{2d}{d} = 2 \]



Step 4: Final Answer:
The ratio $\frac{B(2d)}{B(d)}$ is 2.